An Art and craft teacher prepared a fan which looked like the sector of a circle using ice-cream sticks and black sheet as shown in the given figure: (i) The angle subtended at the centre of the sector is 60°. Find the area of one face of the fan. (ii) With the given sector angle of 60° at the centre, find area of black sheet used in the fan. (iii)(A) The boundary of the black sheet region on one face is to be covered by a designer tape. What length of tape is required? (iii)(B) If 29.6 cm of designer tape is required to cover the boundary of the black sheet region on one face of the fan, then evaluate the central angle of the sector.

MathsCBSE Class 10 · Basic · Sample paper, session 2026-27 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic sample paper 2026-27: question 36

From the official CBSE sample question paper for 2026-27, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedAreas Related to Circles

An Art and craft teacher prepared a fan which looked like the sector of a circle using ice-cream sticks and black sheet as shown in the given figure:

(i)
The angle subtended at the centre of the sector is $6 0^{\circ}$ . Find the area of one face of the fan.
1 mark
(ii)
With the given sector angle of $6 0^{\circ}$ at the centre, find area of black sheet used in the fan.
1 mark
(iii)(A)
The boundary of the black sheet region on one face is to be covered by a designer tape. What length of tape is required?
2 marks
OR
(iii)(B)
If 29.6 cm of designer tape is required to cover the boundary of the black sheet region on one face of the fan, then evaluate the central angle of the sector.
2 marks

Figure in the paper: A folding fan shaped as a sector of a circle. The black sheet forms the outer sector region (fan face) and white ice-cream sticks form the narrow part near the pivot at the bottom. A straight edge is marked 10 cm (the outer radius of the black sheet) and the shorter edge near the pivot is marked 4 cm (the radius of the ice-cream stick portion). Arrows label 'Black sheet' pointing to the dark fabric region and 'Ice-cream sticks' pointing to the white stick region at the bottom. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

Open one hint at a time and try again before the next.

Hint 1

The whole fan face is a sector of radius 10 cm. The black sheet is the part of it outside the sector of radius 4 cm.

Hint 2

Area of a sector is $\frac{θ}{360} π r^{2}$ . For the black region, subtract the small sector from the big one.

Hint 3

The tape goes along two straight edges of length $10 - 4$ each, plus the outer arc and the inner arc, where arc length is $\frac{θ}{360} \times 2 π r$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Given: central angle $θ = 6 0^{\circ}$ , outer radius $R = 10$ cm, radius of the stick portion $r = 4$ cm.
(i) Area of one face $= \frac{θ}{36 0 ^{\circ}} π R^{2} = \frac{60}{360} \times \frac{22}{7} \times 1 0^{2}$
½ mark
(i) $= \frac{1100}{21} \approx 52.38$ cm²
½ mark
(ii) The black sheet is the region between the two arcs, so its area $= \frac{θ}{36 0 ^{\circ}} π (R^{2} - r^{2}) = \frac{60}{360} \times \frac{22}{7} \times (100 - 16)$
½ mark
(ii) $= \frac{1}{6} \times \frac{22}{7} \times 84 = 44$ cm²
½ mark
(iii)(A) The boundary is made of two straight edges of length $(R - r)$ each, the outer arc and the inner arc. Length $= 2 (10 - 4) + \frac{60}{360} \times 2 \times \frac{22}{7} \times (10 + 4)$
1 mark
(iii)(A) $= 12 + \frac{1}{6} \times 88 = 12 + \frac{44}{3} = \frac{80}{3}$ cm $\approx 26.67$ cm of tape
1 mark
(iii)(B) Using the same boundary: $2 (10 - 4) + \frac{θ}{360} \times 2 \times \frac{22}{7} \times (10 + 4) = 29.6$
½ mark
(iii)(B) $12 + \frac{θ}{360} \times 88 = 29.6 \Rightarrow \frac{θ}{360} \times 88 = 17.6$
1 mark
(iii)(B) $θ = \frac{17.6 \times 360}{88} = 7 2^{\circ}$
½ mark

Answer: (i) 52.38 cm²; (ii) 44 cm²; (iii)(A) 80/3 cm ≈ 26.67 cm; (iii)(B) 72°

Where the marks are

(i) Sector area formula with R = 10 cm substituted (½ mark) and result 52.38 cm² (½ mark)
(ii) Area of black sheet as difference of sectors (½ mark) and result 44 cm² (½ mark)
(iii)(A) Boundary expression: two straight edges plus two arcs (1 mark); length 80/3 cm (1 mark)
(iii)(B) Forms equation for the boundary equal to 29.6 (½ mark); simplifies to θ/360 × 88 = 17.6 (1 mark); θ = 72° (½ mark)

Revise the chapter

Chapter: Areas Related to Circles (notes, formulas, where marks are lost and every past question)
Topic: Area of a Sector of a Circle

Practise this chapterBoard-style questions from Areas Related to Circles, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2026-27 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic sample paper 2026-27: question 36

Hints

Step-by-step solution

Revise the chapter

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